With any kind of agreement involving money, you want everything in writing. I mostly agree that the woman's case is highly questionable, and she probably would lose in court. The problem is that once you make an outrageous claim, it's very difficult to back down from it. So I think it will go to court, and she will lose in the end.
That said, everyone involved ought to have put the whole thing in writing, including having a record of the purchased lottery tickets prior to the drawing. But hindsight is 20/20. If people were that on the ball, there would probably be a whole lot fewer lawsuits.